Subgroups — #453×C₃₁₇ #35202808

Order 20288.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	AGL(1,ℤ/8)×C₃₁₇	C₂	1	2	yes	maximal, complemented
3	#127×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#104×C₃₁₇	C₂	1	2	yes	maximal, complemented
5	#100×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₃₁₇	—	2	2	no	—
7	#41×C₃₁₇	C₂²	1	2	yes	—
8	C₂×D₄×C₃₁₇	—	2	2	no	—
9	C₂×D₄×C₃₁₇	—	2	2	no	—
10	C₂×D₄×C₃₁₇	—	2	2	no	—
11	C₂×D₄×C₃₁₇	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₃₁₇	C₂²	1	1	yes	characteristic, complemented
13	SD₁₆×C₃₁₇	—	2	2	no	—
14	D₈×C₃₁₇	—	2	2	no	—
15	M₁₆×C₃₁₇	C₂²	1	2	yes	complemented
16	C₄²×C₃₁₇	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₃₁₇	—	2	2	no	—
18	C₂³×C₃₁₇	—	2	1	no	—
19	C₂³×C₃₁₇	D₄	1	2	yes	—
20	Q₈×C₃₁₇	D₄	1	2	yes	complemented
21	D₄×C₃₁₇	—	2	4	no	—
22	D₄×C₃₁₇	—	2	2	no	—
23	D₄×C₃₁₇	—	2	2	no	—
24	D₄×C₃₁₇	—	2	2	no	—
25	D₄×C₃₁₇	—	4	4	no	—
26	D₄×C₃₁₇	D₄	1	2	yes	complemented
27	C₂×C₄×C₃₁₇	—	2	2	no	—
28	C₂×C₄×C₃₁₇	—	2	1	no	—
29	C₂×C₄×C₃₁₇	—	2	2	no	—
30	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
31	C₈×C₃₁₇	—	2	2	no	—
32	C₂²×C₃₁₇	—	2	2	no	—
33	C₂²×C₃₁₇	—	2	2	no	—
34	C₂²×C₃₁₇	—	4	2	no	—
35	C₂²×C₃₁₇	—	4	2	no	—
36	C₂²×C₃₁₇	—	4	1	no	—
37	C₂²×C₃₁₇	—	8	2	no	—
38	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
39	C₄×C₃₁₇	—	2	2	no	—
40	C₄×C₃₁₇	—	2	2	no	—
41	C₄×C₃₁₇	C₂×D₄	1	2	yes	—
42	C₂×C₃₁₇	—	2	1	no	—
43	C₂×C₃₁₇	—	4	2	no	—
44	C₂×C₃₁₇	—	4	2	no	—
45	C₂×C₃₁₇	—	8	1	no	—
46	C₂×C₃₁₇	#120	1	1	yes	characteristic
47	C₃₁₇	#453	1	1	yes	characteristic, direct product factor, Sylow
48	#453	C₃₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	H(5,𝔽₂)	C₂×C₃₁₇	1	1	yes	characteristic, complemented
50	AGL(1,ℤ/8)	C₂×C₃₁₇	1	2	yes	complemented
51	#127	C₂×C₃₁₇	1	1	yes	characteristic, complemented
52	#104	C₂×C₃₁₇	1	2	yes	complemented
53	#100	C₂×C₃₁₇	1	1	yes	characteristic, complemented
54	#41	—	2	2	no	—
55	#41	C₂²×C₃₁₇	1	2	yes	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	2	2	no	—
59	C₂×D₄	C₂²×C₃₁₇	1	1	yes	characteristic
60	C₂×D₄	C₂²×C₃₁₇	1	1	yes	characteristic, complemented
61	SD₁₆	—	2	2	no	—
62	D₈	—	2	2	no	—
63	M₁₆	C₂²×C₃₁₇	1	2	yes	complemented
64	C₄²	C₂²×C₃₁₇	1	1	yes	characteristic, complemented
65	C₂³	—	2	2	no	—
66	C₂³	—	2	1	no	—
67	C₂³	D₄×C₃₁₇	1	2	yes	—
68	Q₈	D₄×C₃₁₇	1	2	yes	complemented
69	D₄	—	2	4	no	—
70	D₄	—	2	2	no	—
71	D₄	—	2	2	no	—
72	D₄	—	2	2	no	—
73	D₄	—	4	4	no	—
74	D₄	D₄×C₃₁₇	1	2	yes	complemented
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	C₂³×C₃₁₇	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	8	2	no	—
86	C₂²	C₂×D₄×C₃₁₇	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	2	2	no	—
89	C₄	C₂×D₄×C₃₁₇	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	4	2	no	—
93	C₂	—	8	1	no	—
94	C₂	#120×C₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁₇	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃₁₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁₇ ◁ C₂×C₄×C₃₁₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. H(5,𝔽₂)×C₃₁₇	C₂	semidirect
2. AGL(1,ℤ/8)×C₃₁₇	C₂	semidirect
3. #127×C₃₁₇	C₂	semidirect
4. #104×C₃₁₇	C₂	semidirect
5. #100×C₃₁₇	C₂	semidirect
12. C₂×D₄×C₃₁₇	C₂²	semidirect
15. M₁₆×C₃₁₇	C₂²	semidirect
16. C₄²×C₃₁₇	C₂²	semidirect
20. Q₈×C₃₁₇	D₄	semidirect
26. D₄×C₃₁₇	D₄	semidirect
47. C₃₁₇	#453	direct (N × H)
48. #453	C₃₁₇	direct (N × H)
49. H(5,𝔽₂)	C₂×C₃₁₇	semidirect
50. AGL(1,ℤ/8)	C₂×C₃₁₇	semidirect
51. #127	C₂×C₃₁₇	semidirect
52. #104	C₂×C₃₁₇	semidirect
53. #100	C₂×C₃₁₇	semidirect
60. C₂×D₄	C₂²×C₃₁₇	semidirect
63. M₁₆	C₂²×C₃₁₇	semidirect
64. C₄²	C₂²×C₃₁₇	semidirect
68. Q₈	D₄×C₃₁₇	semidirect
74. D₄	D₄×C₃₁₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
H(5,𝔽₂)×C₃₁₇	2
AGL(1,ℤ/8)×C₃₁₇	2
#127×C₃₁₇	2
#104×C₃₁₇	2
#100×C₃₁₇	2
#453	317

Subgroups — #453×C317 #35202808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #453×C₃₁₇ #35202808